International Journal of Engineering Research and Development
e-ISSN: 2278-067X, p-ISSN: 2278-800X, www.ijerd.com
Volume 4, Issue 5 (October 2012), PP. 60-70
Improvement of Dynamic Stability of a Single Machine
Infinite-Bus Power System using Fuzzy Logic based Power
System Stabilizer
Venkatesh Gudla1, P. Kanta Rao2
1
PG Scholar & Department of Electrical and Electronics Engineering, SRKR Engineering College, Bhimavaram-534204.
2
HOD of EEE & GMRIT, Rajam, Srikakulam District-532409.
Abstract:— with constraints on data availability and for study of power system stability it is adequate to model
the synchronous generator with field circuit and one equivalent damper on q-axis known as the model 1.1. This
paper presents a systematic procedure for modelling and simulation of a single-machine infinite-bus power system
installed with a Power System Stabilizer (PSS) and Fuzzy Logic Power System Stabilizer (FLPSS) where the
synchronous generator is represented by model 1.1, so that impact of PSS on power system stability can be more
reasonably evaluated. The model of the example power system is developed using MATLAB/SIMULINK which
can be used for teaching the power system stability phenomena, and also for research works especially to develop
generator controllers using advanced technologies. An analytical approach is developed for the determination of
PSS parameters. The non-linear simulation results are presented to validate the effectiveness of the proposed
approach.
Keywords:— MATLAB/SIMULINK, modelling and simulation, power system stability, single-machine infinitebus power system, Power System Stabilizer, Fuzzy Logic Power System Stabilizer.
NOMENCLATURE
δ
Rotor angle of synchronous generator in radians
ωB
Rotor speed deviation in rad/sec
Sm
Generator slip in p.u.
Smo
Initial operating slip in p.u.
H
Inertia constant
D
Damping coefficient
Tm
Mechanical power input in p.u.
Te
Electrical power output in p.u.
E fd
Excitation system voltage in p.u.
T' do
Open circuit d-axis time constant in sec
T' qo
Open circuit q-axis time constant in sec
xd
d-axis synchronous reactance in p.u.
x' d
d-axis transient reactance in p.u.
xq
q-axis synchronous reactance in p.u.
x' q
q-axis transient reactance in p.u.
I.
INTRODUCTION
Traditionally, for the small signal stability studies of a single-machine infinite-bus (SMIB) power system, the
linear model of Phillips-Heffron has been used for years, providing reliable results [1]-[2]. It has also been successfully used
for designing and tuning the classical power system stabilizers (PSS). Although the model is a linear model, it is quite
accurate for studying low frequency oscillations and stability of power systems. With the advent of Flexible AC
Transmission System (FACTS) devices [3], such as Thyristor Controlled Series Compensator (TCSC), Static synchronous
compensator (STATCOM) and unified power flow controller (UPFC), the unified model of SMIB power system installed
with a TCSC, STATCOM and a UPFC have been developed [4]-[6]. These models are the popular tools amongst power
engineers for studying the dynamic behaviour of synchronous generators, with a view to design control equipment. However,
the model only takes into account the generator main field winding and hence these models may not always yield a realistic
dynamic assessment of the SMIB power system with FACTS because the generator damping winding in q-axis is not
accounted for. Further, liner methods cannot properly capture complex dynamics of the system, especially during major
disturbances. This presents difficulties for designing the FACTS controllers in that, the controllers designed to provide
desired performance at small signal condition do not guarantee acceptable performance in the event of major disturbances.
Effective design and accurate evaluation of the PSS control strategy depend on the accuracy of modelling of this
process. In [7], a systematic procedure for modelling, simulation and optimal tuning of PSS controller in a SMIB power
system was presented where the MATLAB/SIMULINK based model was developed to design the PSS. However, the model
only takes into account the generator main field winding and the synchronous machine was represented by model (1.0). This
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Improvement of Dynamic Stability of a Single Machine Infinite-Bus Power System using Fuzzy..
paper presents a higher-order synchronous machine model, which includes one damper winding along the q-axis, for a power
system installed with a PSS and FLPSS.
Despite significant strides in the development of advanced control schemes over the past two decades, the
conventional lead-lag (LL) structure controller as well as the classical proportional-integral-derivative (PID) controller and
its variants, remain the controllers of choice in many industrial applications. These controller structures remain an engineer’s
preferred choice because of their structural simplicity, reliability, and the favourable ratio between performance and cost.
Beyond these benefits, these controllers also offer simplified dynamic modelling, lower user-skill requirements, and minimal
development effort, which are issues of substantial importance to engineering practice [8]-[9]. In [10], a comparative study
about the PSS and FLPSS based design was presented, where it has been shown that that LL structured PSS with the
controller parameters, gives the best system response compared to all other alternatives. In view of the above, a LL
controller structure is used for the PSS controller.
The problem of PSS parameter tuning is a complex exercise. A number of conventional techniques have been
reported in the literature pertaining to design problems of conventional power system stabilizers namely: the eigenvalue
assignment, mathematical programming, gradient procedure for optimization and also the modern control theory.
Unfortunately, the conventional techniques are time consuming as they are iterative and require heavy computation burden
and slow convergence. In addition, the search process is susceptible to be trapped in local minima and the solution obtained
may not be optimal [11].
This paper is organized as follows. In Section II, the modelling of power system under study, which is a SMIB
power system with a PSS and FLPSS, is presented. The proposed controller structures and problem formulation are
described in Section III. Simulation results are provided and discussed in Section IV and conclusions are given in Section V
II.
POWER SYSTEM UNDER STUDY
The SMIB power system shown in Fig. 1 is considered in this study. The synchronous generator is delivering
power to the infinite-bus through a double circuit transmission line. In Fig. 1, Vt and Eb are the generator terminal and
infinite bus voltage respectively; XT , X L and XTH represent the reactance of the transformer, transmission line per circuit and
the Thevenin’s impedance of the receiving end system respectively.
Vt ∟θ
Eb∟0
jXL
jXTH
jXT
jXL
Fig. 1 Single-machine infinite-bus power system
A. Modelling the Synchronous Generator Infinite-bus Power System
The synchronous generator is represented by model 1.1, i.e. with field circuit and one equivalent damper winding on q-axis.
The machine equations are [12]:
𝑑𝛿
= 𝜔𝐵 ( 𝑆𝑚 − 𝑆𝑚 0 )
𝑑𝑡
→ (1)
𝑑𝑆𝑚
1
=
[−𝐷( 𝑆𝑚 − 𝑆𝑚 0 ) + 𝑇𝑚 − 𝑇𝑒 ]
𝑑𝑡
2𝐻
→ (2)
𝑑𝐸′𝑞
1
=
[−𝐸 ′ 𝑞 + ( 𝑥𝑑 − 𝑥 ′ 𝑑 )𝑖𝑑 + 𝐸𝑓𝑑 ]
𝑑𝑡
𝑇′𝑑0
→ (3)
𝑑𝐸′𝑑
1
=
[−𝐸 ′ 𝑑 + ( 𝑥𝑞 − 𝑥 ′ 𝑞 )𝑖𝑞 ]
𝑑𝑡
𝑇′𝑞0
The electrical torque Te is expressed in terms of variables E' d , E' q , id and iq as:
𝑇𝑒 = 𝐸 ′ 𝑑 𝑖𝑑 + 𝐸 ′ 𝑖𝑞 + ( 𝑥′𝑑 − 𝑥′𝑞 )𝑖𝑑 𝑖𝑞
𝑞
→ (4)
→ (5)
For a lossless network, the stator algebraic equations and the network equations are expressed as:
𝐸 ′ 𝑞 + 𝑥′𝑑 𝑖𝑑 = 𝑣𝑞
→ (6)
𝐸 ′ 𝑑 − 𝑥′𝑞 𝑖𝑞 = 𝑣𝑑
→ (7)
𝑣𝑞 = − 𝑥𝑒 𝑖𝑑 + 𝐸𝑏 𝑐𝑜𝑠 𝛿
→ (8)
𝑣𝑑 = 𝑥𝑒 𝑖𝑞 − 𝐸𝑏 𝑠𝑖𝑛 𝛿
→ (9)
Solving the above equations, the variables id and iq can be obtained as:
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Improvement of Dynamic Stability of a Single Machine Infinite-Bus Power System using Fuzzy..
𝑖𝑑 =
𝐸𝑏 𝑐𝑜𝑠 𝛿 − 𝐸 ′ 𝑞
𝑥𝑒 + 𝑥′𝑑
→ (10)
𝑖𝑞 =
𝐸𝑏 𝑠𝑖𝑛 𝛿 − 𝐸 ′ 𝑑
𝑥𝑒 + 𝑥′𝑞
→ (11)
The above notation for the variables and parameters described are standard and defined in the nomenclature. For more
details, the readers are suggested to refer [11]-[12].
III.
PROBLEM FORMULATION
A. Structure of the PSS
The structure of PSS-based damping controller is shown in Fig. 2. The input signal of the proposed controllers is the speed
deviation (Δω), and the output signal is the VPSS . The structure consists of a gain block with gain KPSS, a signal washout
block and two-stage phase compensation blocks. The signal washout block serves as a high-pass filter, with the time
constant TW, high enough to allow signals associated with oscillations in input signal to pass unchanged.
Fig. 2 Structure of PSS
From the viewpoint of the washout function, the value of TW is not critical and may be in the range of 1 to 20
seconds [12]. The phase compensation block (time constants T1, T2 and T3, T4) provides the appropriate phase-lead
characteristics to compensate for the phase lag between input and the output signals.
The block diagram of the PSS used in industry is shown in Fig 2. It consists of a washout circuit, dynamic
compensator, torsion filter and limiter. PSS with guidelines for the selection of parameters (tuning) are given next.
It is to be noted that the major objectives of providing PSS is to increase the power transfer in the network, which would
otherwise be limited by oscillatory instability. The PSS must also function properly when the system is subjected to large
disturbances.
PSS have been used for over 20 years in Western systems of United States of America and in Ontario Hydro. In
United Kingdom, PSS have been used in Scotland to damp oscillations in tie lines connecting Scotland and England. It can
be generally said that need for PSS will be felt in situations when power has to be transmitted over long distances with weak
AC ties. Even when PSS may not be required under normal operating conditions, they allow satisfactory operation under
unusual or abnormal conditions which may be encountered at times. Thus, PSS has become a standard option with modern
static exciters and it is essential for power engineers to use these effectively. Retrofitting of existing excitation systems with
PSS may also be required to improve system stability.
B.
PSS Design
The tuning of Power System Stabilizer can be performed using extensive analytical studies covering various
aspects. While such studies are useful in optimizing the performance of PSS, satisfactory operation of PSS can be obtained
by tuning the PSS.
1. Measure the open loop frequency response without PSS. This involves obtaining the transfer function between the
terminal voltage and the AVR input (Vs) in frequency domain. As described earlier, the transfer function is
approximately related to GEP(s).
2. Select PSS time constants by trial and error such that desired phase compensation is obtained. The guidelines for
selecting the phase compensation are:
a. Check that the compensated system (GEP(s) PSS(s)) has some phase lag at inter area modes.
b. Verify the stabilizer time constant settings by field test which involves determination of points on a root locus. The
local mode oscillations are stimulated by step changes to AVR reference, line switching or low level sinusoidal
modulation (at local mode frequency) of the voltage reference. The effect of the PSS can be measured by
comparing the damping with zero PSS gain and few low values of the gain which cause a noticeable change. The
waveform recorded can give information on the frequency and damping ratio.
3. Perform the gain margin test. This consists of slowly increasing the stability gain until instability is observed
which is characterized by growing oscillations at a frequency greater than the local mode. The oscillation can be
monitored from PSS output. Once instability is detected the stabilizer is switched out of service. Reduction of
stabilizer output limits during the test will ensure that safe operation of the generator is maintained.
4. The PSS gain can now be set to a lower value and a fraction of the instability gain. Typically the gain is set to 1/3
of the instability gain.
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Improvement of Dynamic Stability of a Single Machine Infinite-Bus Power System using Fuzzy..
C.
Fuzzy Logic PSS
The model of FLPSS refers on fuzzy logic controller block and to simplify model integrator block is also removed
FLPSS model is as shown in Fig. 3
Fig. 3 FLPSS model
Fig. 3 shows that FLPSS consists of blocks, those are wash-out, gain (Kw, Kp, Ks), and limiter. Each part is explained as
follows:
1. Wash Out
This block consists of two wash-out filter with time constant 10 second.
2. Gain [Kw Kp Ks]
Gain is needed to normalize the input and output of fuzzy logic controller.
3. Fuzzy Logic Controller
Fuzzy Logic Controller has a function to produce control signal as an output appropriate with the input.
4. Limiter [Vsmin Vsmax]
Limiter gives limitation of the PSS output.
In order to set fuzzy logic controller, input membership function, output membership function, rules, and gain tuning are
definitive. Detail of input membership function, output membership function, rules, and gain tuning are as follows:
Input Membership Function
Each of input variables are classified by Input membership function consists of trapezoidal membership function and
triangular membership function, described in Fig. 4.
Fig. 4 Input Membership Function
Output Membership Function
Output membership function also consists of triangular membership function, described in Fig. 5.
Fig. 5 Output Membership Function
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Improvement of Dynamic Stability of a Single Machine Infinite-Bus Power System using Fuzzy..
Rules
Rules used by FLPSS are shown in Table I.
dw/dpa
BN
MN
LN
Z
LP
MP
BP
Table I: Fuzzy Logic Rules
MP
LP
Z
LN
LN
MN
MN
BN
BZ
LN
MN
MN
LP
BZ
LN
LN
MP
LP
LZ
LN
MP
LP
LP
BZ
BP
MP
MP
LP
BP
BP
MP
MP
BP
BZ
LP
MP
BP
BP
BP
BP
MN
BN
BN
MN
MN
LN
BZ
LP
BN
BN
BN
BN
BN
MN
LN
BZ
Gain Tuning
The gains of the proportional and derivative actions of the FLPSS are given by the following relations:
𝐾𝑃𝑅 = 𝐾𝑆 × 𝐹 𝐾𝑃 + 𝐹 𝐾𝑊
→ (12)
𝐾𝐷𝐸𝑅 = 𝐾𝑆 × 𝐹 𝐾𝑃
→ (13)
With KPR, KDER, and F{} are proportional gain, derivative gain, and fuzzy operation, respectively. To obtain the gains Kw, Kp,
and Ks a two steps method has been used. These two steps consist of adjusting K p and Kw in order to normalize input and
then tuning Ks to obtain best result. The value of Kw, Kp, and Ks are 0.55, 50 x Kp, and 3, respectively.
D.
Problem Formulation
In the present study, a washout time constant of TW=10s is used. The controller gain KPSS and the time constants
T1, T2, T3 and T4 are to be determined.
IV.
RESULTS AND DISCUSSIONS
In order to optimally tune the parameters of the PSS and FLPSS, as well as to assess its performance and
robustness under wide range of operating conditions with various fault disturbances and fault clearing sequences, the
MATLAB/SIMULINK model of the example power system is developed using equations (2)–(11).
The objective function is evaluated for each individual by simulating the system dynamic model considering a three phase
fault at the generator terminal busbar at t = 20 sec.
Simulation Results
In order to show the advantages of modelling the synchronous generator with PSS and tuning its Parameters in the
way presented in this paper, simulation studies are carried out for the example power system subjected to various severe
disturbances as well as small disturbance. The following cases are considered:
Case-1: Three-phase Fault Disturbance
A three phase fault is applied at the generator terminal busbar at t = 20 sec and cleared after 5 cycles. The original
system is restored upon the fault clearance. To study the performance of PSS controller, two cases are considered; with and
without PSS and FLPSS. The response without the controller (no control) is shown with blue line with legend NC, the
responses with PSS is shown with green line with legend PSS and the responses with FLPSS is shown with red line with
legend FLPSS.
32.5
NC
PSS
FLPSS
32.4
Delta(deg)
32.3
32.2
32.1
32
31.9
0
10
20
30
40
50
Time(sec)
60
70
80
90
100
Fig. 6 Variation of power angle δ, without, with PSS and FLPSS for a 5-cycle three-phase fault disturbance (Case-1)
The system power angle response for the above contingency is shown in Fig. 6. It is clear from the Fig. 6 that,
without controller even though the system is stable, power system oscillations are poorly damped.
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Improvement of Dynamic Stability of a Single Machine Infinite-Bus Power System using Fuzzy..
It is also clear that, proposed PSS significantly suppresses the oscillations in the power angle and provides good damping
characteristics to low frequency oscillations by stabilizing the system faster and the controller FLPSS is further damped
oscillations by stabilizing the system much faster. Figs. 7 - 14 shows the variation of speed deviation Δω, electrical power
Pe , voltages E' d , E' q ,E fd , Vt , currents id and iq , respectively all with respect to time for the above mentioned
contingency (Case-1).
It is clear from this figure that, the PSS and FLPSS improves the stability performance of the example power system and
power system oscillations are well damped out.
0.02
NC
PSS
FLPSS
0.015
0.01
Dw(pu)
0.005
0
-0.005
-0.01
-0.015
0
10
20
30
40
50
Time(sec)
60
70
80
90
100
Fig. 7 Variation of speed deviation Δω : Case-1
1
0.9
NC
PSS
FLPSS
0.8
0.7
P(pu)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
10
20
30
40
50
Time(sec)
60
70
80
90
100
Fig. 8 Variation of electrical power Pe : Case-1
-0.1
-0.11
NC
PSS
FLPSS
-0.12
-0.13
Ed'(pu)
-0.14
-0.15
-0.16
-0.17
-0.18
-0.19
-0.2
0
10
20
30
40
50
Time(sec)
60
70
80
90
100
Fig. 9 Variation of voltage E' d : Case-1
1.5
NC
PSS
FLPSS
1.45
Eq'(pu)
1.4
1.35
1.3
1.25
0
10
20
30
40
50
Time(sec)
60
70
80
Fig. 10 Variation of voltage E' q : Case-1
65
90
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Improvement of Dynamic Stability of a Single Machine Infinite-Bus Power System using Fuzzy..
6
NC
PSS
FLPSS
5
4
Ef(pu)
3
2
1
0
-1
-2
0
10
20
30
40
50
Time(sec)
60
70
80
90
100
Fig. 11 Variation of voltage Ef : Case-1
1.25
NC
PSS
FLPSS
1.2
Vt(pu)
1.15
1.1
1.05
1
0
10
20
30
40
50
Time(sec)
60
70
80
90
100
Fig. 12 Variation of terminal voltage Vt : Case-1
-0.3
NC
PSS
FLPSS
-0.4
Id(pu)
-0.5
-0.6
-0.7
-0.8
-0.9
0
10
20
30
40
50
Time(sec)
60
70
80
90
100
Fig. 13 Variation of current Id : Case-1
0.45
NC
PSS
FLPSS
0.4
Iq(pu)
0.35
0.3
0.25
0.2
0.15
0
10
20
30
40
50
Time(sec)
60
70
80
90
100
Fig. 14 Variation of current Iq : Case-1
Case-2: Line-outage Disturbance
In this case another severe disturbance is considered. One of the transmission line is permanently tripped out at t = 20 sec.
The system response for the above contingency is shown in Figs. 15- 23.
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Improvement of Dynamic Stability of a Single Machine Infinite-Bus Power System using Fuzzy..
32.4
NC
PSS
FLPSS
32.35
Delta(pu)
32.3
32.25
32.2
32.15
0
10
20
30
40
50
Time(sec)
60
70
80
90
100
Fig. 15 Variation of power angle δ : Case-2
-3
5
x 10
4
NC
PSS
FLPSS
3
2
Dw(pu)
1
0
-1
-2
-3
-4
0
10
20
30
40
50
Time(sec)
60
70
80
90
100
Fig. 16 Variation of speed deviation Δω : Case-2
0.68
0.66
NC
PSS
FLPSS
0.64
0.62
P(pu)
0.6
0.58
0.56
0.54
0.52
0.5
0
10
20
30
40
50
Time(sec)
60
70
80
90
100
Fig. 17 Variation of electrical power Pe : Case-2
-0.16
-0.162
NC
PSS
FLPSS
-0.164
-0.166
Ed'(pu)
-0.168
-0.17
-0.172
-0.174
-0.176
-0.178
-0.18
0
10
20
30
40
50
Time(sec)
60
70
80
Fig. 18 Variation of voltage E' d : Case-2
67
90
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Improvement of Dynamic Stability of a Single Machine Infinite-Bus Power System using Fuzzy..
1.34
NC
PSS
FLPSS
1.33
Eq'(pu)
1.32
1.31
1.3
1.29
1.28
0
10
20
30
40
50
Time(sec)
60
70
80
90
100
Fig. 19 Variation of voltage E' q : Case-2
3
NC
PSS
FLPSS
2.8
2.6
Ef(pu)
2.4
2.2
2
1.8
1.6
1.4
0
10
20
30
40
50
Time(sec)
60
70
80
90
100
Fig. 20 Variation of voltage Ef : Case-2
1.22
NC
PSS
FLPSS
1.215
1.21
Vt(pu)
1.205
1.2
1.195
1.19
1.185
1.18
0
10
20
30
40
50
Time(sec)
60
70
80
90
100
Fig. 21 Variation of terminal voltageVt : Case-2
-0.44
NC
PSS
FLPSS
-0.46
-0.48
Id(pu)
-0.5
-0.52
-0.54
-0.56
-0.58
-0.6
0
10
20
30
40
50
Time(sec)
60
70
80
Fig. 22 Variation of line current Id: Case-2
68
90
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Improvement of Dynamic Stability of a Single Machine Infinite-Bus Power System using Fuzzy..
0.34
NC
PSS
FLPSS
0.33
0.32
Iq(pu)
0.31
0.3
0.29
0.28
0.27
0
10
20
30
40
50
Time(sec)
60
70
80
90
100
Fig. 23 Variation of line current Iq: Case-2
It is also clear from the Figs. that he PSS has good damping characteristics to low frequency oscillations and
quickly stabilizes the system under this severe disturbance.
Case-3: Small Disturbance
In order to verify the effectiveness of PSS under small disturbance, the mechanical power input to the generator is
decreased by 1 pu at t = 20 sec and the disturbance is removed at t = 50 sec. The system response under this small
disturbance contingency is shown in Figs. 24-27.
32.25
NC
PSS
FLPSS
32.2
Delta(deg)
32.15
32.1
32.05
32
0
10
20
30
40
50
Time(sec)
60
70
80
90
100
Fig. 24 Variation of power angle δ: Case-3
-3
4
x 10
NC
PSS
FLPSS
3
2
Dw(pu)
1
0
-1
-2
-3
-4
0
10
20
30
40
50
Time(sec)
60
70
80
90
100
Fig. 25 Variation of speed deviation Δω : Case-3
0.66
0.64
NC
PSS
FLPSS
0.62
0.6
P(pu)
0.58
0.56
0.54
0.52
0.5
0.48
0.46
0
10
20
30
40
50
Time(sec)
60
70
80
Fig. 26 Variation of electrical power Pe : Case-3
69
90
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Improvement of Dynamic Stability of a Single Machine Infinite-Bus Power System using Fuzzy..
1.21
Vt(pu)
1.205
1.2
1.195
1.19
1.185
0
10
20
30
40
50
Time(sec)
60
70
80
90
100
Fig. 27 Variation of terminal voltageVt : Case-3
It is clear from the Figs. 21-24 that, the PSS has good damping characteristics to low frequency oscillations and
quickly stabilizes the system under this small disturbance.
V.
CONCLUSIONS
The MATLAB/SIMULINK model of a single-machine infinite-bus power system with a PSS and FLPSS
controller presented in the paper provides a means for carrying out power system stability analysis and for explaining the
generator dynamic behavior as affected by a PSS. This model is far more realistic compared to the model available in open
literature, since the synchronous generator with field circuit and one equivalent damper on q-axis is considered. The
controller is tested on example power system subjected to various large and small disturbances. The simulation results show
that, the PSS and FLPSS improves the stability performance of the power system and power system oscillations are
effectively damped out. Hence, it is concluded that the proposed model is suitable for carrying out power system stability
studies in cases where the dynamic interactions of a synchronous generator of a PSS and FLPSS are the main concern.
APPENDIX
System data: All data are in pu unless specified otherwise.
Generator: H = 3.542, D = 0, Xd=1.7572, Xq=1.5845, X’d =0.4245, X’q =1.04, T’do = 6.66, T’qo=0.44, Ra=0, Pe=0.6,
Qe=0.02224, δ0=44.370. Exciter: KA=400, TA=0.025 s
Transmission line: R=0, XL = 0.8125, XT=0.1364.
PSS: Wash-out network: Ka=15, Tw=10, Lead-lag network: T1=0.75, T2=0.3, Lag-lead network: T3=0.75, T4=0.3,
Vs=±0.05.
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